No changes to LWP required. You can get LWP::Simple to export the underlying LWP::UserAgent object. Once you have that you simply set the proxy option if the environment dictates. Just patch your module like this and it should work:

use LWP::Simple qw( get $ua );

$ua->proxy( ['http'], $ENV{HTTP_PROXY} ) if exists $ENV{HTTP_PROXY};
[download]